biLGP | R Documentation |
Density, random-number generation, and moments of the log-transformed distribution.
dbiLGP(y, theta, lambda, nc=NULL, log=FALSE, add.carefully=FALSE) biLGP.logMV(theta,lambda,nc=NULL,const.add=1,tol=1e-14,add.carefully=FALSE) rbiLGP(n, theta, lambda)
y |
Numeric vector or two-column matrix of bivariate data. If matrix, each row corresponds to an observation. |
theta |
Numeric vector or three-column matrix of non-negative values for index parameters theta0, theta1, and theta2, in that order. If matrix, is read by row. |
lambda |
Numeric vector or three-column matrix of values for multiplicative parameters lambda0, lambda1, and lambda2, in that order. If matrix, is read by row. Values must be on the interval [-1,1]. |
nc |
Numeric vector or three-column matrix of (reciprocals of) the normalizing constants. These constants differ from 1 only if the corresponding |
log |
Logical; should the natural log of the probability be returned? Defaults to |
add.carefully |
Logical. If |
const.add |
Numeric vector of positive constants to add to the non-negative integers before taking their natural logarithm. Defaults to 1, for the typical log(y+1) transformation. |
tol |
Numeric; must be positive. When |
n |
Integer; number of observations to be randomly generated. |
The bivariate LGP is constructed from three independent latent random variables, X0, X1, and X2, where
X0 ~ LGP(theta0, lambda0)
X1 ~ LGP(theta1, lambda1)
X2 ~ LGP(theta2, lambda2)
The observable variables, Y1 and Y2, are defined as Y1 = X0 + X1 and Y2 = X0 + X2, and thus the dependence between Y1 and Y2 arises because of the common term X0. The joint PMF of Y1 and Y2 is derived from the joint PMF of the three independent latent variables, with X1 and X2 re-expressed as Y1 - X0 and Y2 - X0, and after X0 is marginalized out.
Function dbiLGP()
is the bivariate LGP density (PMF). Function rbiLGP()
generates random draws from the bivariate LGP distribution, via calls to rLGP()
. Function biLGP.logMV()
numerically computes the means, variances, and covariance of a bivariate LGP distribution, after it has been log transformed following addition of a positive constant.
Vectors of numeric arguments other than tol
are cycled, whereas only the first element of logical and integer arguments is used.
dbiLGP()
returns a numeric vector of probabilities. rbiLGP()
returns a matrix of random draws, which is of type 'numeric' (rather than 'integer', even though the bivariate LGP only has support on the non-negative integers). biLGP.logMV()
returns a numeric matrix with the following five named columns:
EY1
: Post-tranformation expectation of Y1.
EY2
: Post-tranformation expectation of Y2.
VY1
: Post-tranformation variance of Y1.
VY2
: Post-tranformation variance of Y2.
COV
: Post-tranformation covariance of Y1 and Y2.
Robert M. Kirkpatrick rkirkpatrick2@vcu.edu
Famoye, F., & Consul, P. C. (1995). Bivariate generalized Poisson distribution with some applications. Metrika, 42, 127-138.
Consul, P. C., & Famoye, F. (2006). Lagrangian Probability Distributions. Boston: Birkhauser.
LGP
, dpois()
## The following two lines do the same thing: dbiLGP(y=1,theta=1,lambda=0.1) dbiLGP(y=c(1,1),theta=c(1,1,1),lambda=c(0.1,0.1,0.1)) dbiLGP(y=c(1,1,2,2,3,5),theta=c(1,1,1,2,2,2),lambda=0.1) ## Due to argument cycling, the above line is doing the following three steps: dbiLGP(y=c(1,1),theta=c(1,1,1),lambda=c(0.1,0.1,0.1)) dbiLGP(y=c(2,2),theta=c(2,2,2),lambda=c(0.1,0.1,0.1)) dbiLGP(y=c(3,5),theta=c(1,1,1),lambda=c(0.1,0.1,0.1)) ## Inputs to dbiLGP() can be matrices, too: dbiLGP(y=matrix(c(1,1,2,2,3,5),ncol=2,byrow=TRUE), theta=matrix(c(1,1,1,2,2,2,1,1,1),ncol=3,byrow=TRUE), lambda=0.1) ## theta0 = 0 implies independence: a <- dbiLGP(y=c(1,3),theta=c(0,1,2),lambda=c(0.1,-0.1,0.5)) b <- dLGP(x=1,theta=1,lambda=-0.1) * dLGP(x=3,theta=2,lambda=0.5) a-b #<--near zero. ## lambdas of zero yield the ordinary Poisson: a <- dbiLGP(y=c(1,3), theta=c(0,1,2),lambda=0) b <- dpois(x=1,lambda=1) * dpois(x=3,lambda=2) #<--LGP theta is pois lambda a-b #<--near zero ( y <- rbiLGP(10,theta=c(1.1,0.87,5.5),lambda=c(0.87,0.89,0.90)) ) dbiLGP(y=y,theta=c(1.1,0.87,5.5),lambda=c(0.87,0.89,0.90)) biLGP.logMV(theta=c(0.65,0.35,0.35),lambda=0.7,tol=1e-8)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.